31

This article is about a particular natural number.|View all articles on particular natural numbers

Summary

Factorization

Properties and families

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
prime number	11th prime number	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, [SHOW MORE] 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271 View list on OEIS	A natural number $n>1$ is prime if and only if $n$ is not divisible by any prime less than or equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{n}} . In this case, since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{31}} is between 5 and 6, verifying primality requires checking that 31 i not divisible by any prime up to 5, i.e., it is not divisible by 2, 3, or 5.
Mersenne number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_n = 2^n - 1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 5} , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_5 = 2^5 - 1}		plug and check
Mersenne prime (both a Mersenne number and a prime number)
regular prime	10th regular prime (note that 2 is neither a regular nor an irregular prime)	3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, [SHOW MORE] 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281, 313, 317, 331, 337, 349, 359, 367, 373, 383, 397, 419, 431 View list on OEIS
Euclid number	number that is of the form primorial + 1	2, 3, 7, 31, 211, 2311, View list on OEIS	30 is a primorial: it is the product of the first three primes

Prime-generating polynomials

Below are some polynomials that give prime numbers for small input values, which give the value 31 for suitable input choice.

Polynomial	Degree	Some values for which it generates primes	Input value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} at which it generates 31
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^2 - n + 11}	2	all numbers 1-10, because 11 is one of the lucky numbers of Euler.	5
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n^2 + 29}	2	all numbers 0-28	1

Multiples

Interesting multiples

Number	Prime factorization	What's interesting about it
341	11 times 31	smallest Poulet number (also called Sarrus number), i.e., smallest Fermat pseudoprime to base 2
2821	7 times 13 times 31	one of the Carmichael numbers, i.e., absolute pseudoprimes